A note on variational representation for singular values of matrix

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1 Alied Mathematics and Comutation 43 (2003) A note on variational reresentation for singular values of matrix Zhi-Hao Cao *, Li-Hong Feng Deartment of Mathematics and Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai , China Abstract In this note we address the variational roerty of singular values of matrix and oint out that a theorem in [Matrix Comutations, John Hokins University Press, Baltimore, MD, 989,993,996] is incomlete. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Singular value Symmetric matrix Saddle oint roblem Let A 2 R mn and let r i ðaþ denote the ith largest singular value of A. The following theorem is given in [2] (Theorem 8.3. in eds. 989 and 993, and Theorem 8.6. in ed. 996): Theorem. If A 2 R mn, then for k : fm ng r k ðaþ ðþ r k ðaþ 2 ð2þ where S R n and T R m are subsaces. * Corresonding author. address: zcao@fudan.edu.cn (Z.-H. Cao) /02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. doi:0.06/s (02)

2 560 Z.-H. Cao, L.-H. Feng / Al. Math. Comut. 43(2003) However, the first art of Theorem, i.e. (), is incomlete, as the following simle examle shows. Examle. Let A =2 then we have, obviously, r 2 ðaþ =2. On the other hand, let ^x ^y 2 then 6 xy2r 2 x ^y T A^x k^xk 2 k^yk 2 0 since ^y T A^x 0. In fact, a variational reresentation for singular values of matrix can be given as follows. Theorem 2. If A 2 R mn, then for k : fm ng r k ðaþ : ð3þ ð4þ Proof. We note that the second art of Theorem, i.e. (2), is correct (see also [3]): r k ðaþ 2 : ð5þ

3 First, let us rove (3), we have Then, let us rove (4), we have r k ðaþ: ðaxþ T y kðaxþ T k 2 r k ða T Þr k ðaþ: ðy T AÞx ky T Ak 2 2 ka T yk 2 Without lose of generality, we assume that m 6 n, then we have the following corollary of Theorem 2. Corollary 3. Let A 2 R mn, m 6 n, then r r m where the singular values of A are ordered as r P r 2 P P r m : The aroximation of saddle oint roblem, for examle, the mixed finite element solution of the Stokes equations describing slow incomressible viscous flow leads to a symmetric indefinite discrete system: A B T u f ð6þ B 0 0 for the ressure comonent and velocity comonent u, where A 2 R nn is symmetric ositive definite, B 2 R mn, m 6 n, is a matrix with full row rank. The matrix form of the so called Babuska Brezzi condition is as follows (cf. []) u60 Z.-H. Cao, L.-H. Feng / Al. Math. Comut. 43(2003) T Bu ðu T AuÞ P =2 cðt M Þ =2 8 2 R m 6 0 ð7þ

4 562 Z.-H. Cao, L.-H. Feng / Al. Math. Comut. 43(2003) where (ressure mass matrix) M 2 R mm is symmetric ositive definite, c is a scalar. Using Corollary 3 we can easily derive the following result (cf. []). Theorem 4. The uer bound of the scalars fcg which satisfy (7) is the smallest singular value r ðm =2 BA =2 Þ of matrix M =2 BA =2 and the Babuska Brezzi condition (7) can be exressed as T ðba B T Þ T M P c R m 6 0: ð8þ Proof. For u 6 0 and 6 0 we have BA =2^u T Bu ðu T AuÞ =2 ð T M Þ ^T =2 M =2 k^uk 2 k^k 2 where ^u A =2 u and ^ M =2. Corollary 3 imlies 2R m 60 u60 T Bu ðu T AuÞ =2 ð T =2 M Þ Comaring (7) and (9) we have Since ^2R m ^60 ^ ^u60 b T ðm =2 BA =2 Þbu k^uk 2 k^k 2 r ðm =2 BA =2 Þ: ð9þ c 6 r ðm =2 BA =2 Þ: ð0þ r 2 ðm =2 BA =2 Þk ðm =2 BA B T M =2 Þ (8) follows from Courant Fisher Mini Theorem [2] and (0). Remark. The bound condition is defined as [] u60 T Bu ðu T AuÞ 6 =2 CðT M Þ =2 8 2 R m 6 0: ðþ Using Corollary 3 we can analogously deduce that the lower bound of the scalars fcg which satisfy () is the largest singular value r ðm =2 BA =2 Þ of matrix M =2 BA =2 and the condition () can be exressed as C 2 P T ðba B T Þ 8 2 R m 6 0: T M

5 Acknowledgements Z.-H. Cao, L.-H. Feng / Al. Math. Comut. 43(2003) This work is suorted by NSFC Project 0702, the Foundation of National Key Laboratory of Comutational Physics and the Doctoral Point Foundation of China. References [] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Sringer-Verlag, New York, 99. [2] G.H. Golub, C.F. Van Loan, Matrix Comutations, John Hokins University Press, Baltimore, MD, 989, 993, 996. [3] G.W. Stewart, Introduction to Matrix Comutations, Academic Press, New York, 973.